What is the general solution for 2sin(2x)+6sin(x)-2cos(x)=3 ?

The general solution for 2sin(2x)+6sin(x)-2cos(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin

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2sin(2x)+6sin(x)-2cos(x)=3

Solution

2 sin (2 x) + 6 sin (x) - 2 cos (x) = 3

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (2 x) + 6 sin (x) - 2 cos (x) = 3

Subtract

3

from both sides

2 sin (2 x) + 6 sin (x) - 2 cos (x) - 3 = 0

Rewrite using trig identities

- 3 - 2 cos (x) + 2 sin (2 x) + 6 sin (x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= - 3 - 2 cos (x) + 2 \cdot 2 sin (x) cos (x) + 6 sin (x)

Simplify

= - 3 - 2 cos (x) + 4 sin (x) cos (x) + 6 sin (x)

- 3 - 2 cos (x) + 6 sin (x) + 4 cos (x) sin (x) = 0

Factor

- 3 - 2 cos (x) + 6 sin (x) + 4 cos (x) sin (x) : (2 cos (x) + 3) (2 sin (x) - 1)

- 3 - 2 cos (x) + 6 sin (x) + 4 cos (x) sin (x)

= (- 2 cos (x) - 3) + (4 sin (x) cos (x) + 6 sin (x))

Factor out

2 sin (x)

from

4 sin (x) cos (x) + 6 sin (x) : 2 sin (x) (2 cos (x) + 3)

4 sin (x) cos (x) + 6 sin (x)

Rewrite

6

3 \cdot 2

Rewrite

4

2 \cdot 2

= 2 \cdot 2 sin (x) cos (x) + 3 \cdot 2 sin (x)

Factor out common term

2 sin (x)

= 2 sin (x) (2 cos (x) + 3)

Factor out

- 1

from

- 2 cos (x) - 3 : - (2 cos (x) + 3)

- 2 cos (x) - 3

Factor out common term

- 1

= - (2 cos (x) + 3)

= 2 sin (x) (2 cos (x) + 3) - (2 cos (x) + 3)

Factor out common term

2 cos (x) + 3

= (2 cos (x) + 3) (2 sin (x) - 1)

(2 cos (x) + 3) (2 sin (x) - 1) = 0

Solving each part separately

2 cos (x) + 3 = 0 or 2 sin (x) - 1 = 0

2 cos (x) + 3 = 0 :

No Solution

2 cos (x) + 3 = 0

Move

3

to the right side

2 cos (x) + 3 = 0

Subtract

3

from both sides

2 cos (x) + 3 - 3 = 0 - 3

Simplify

2 cos (x) = - 3

2 cos (x) = - 3

Divide both sides by

2

2 cos (x) = - 3

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{- 3}{2}

Simplify

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

- 1 \leq cos (x) \leq 1

No Solution

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 sin (x) - 1 = 0

Move

1

to the right side

2 sin (x) - 1 = 0

Add

1

to both sides

2 sin (x) - 1 + 1 = 0 + 1

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Popular Examples

sin(2x)= 4/5

sin (2 x) = \frac{4}{5}

2csc(x)+17=15+csc(x)

2 csc (x) + 17 = 15 + csc (x)

cos(x)-sin(x)=sqrt(2)

cos (x) - sin (x) = 2

cot(θ)=(5.8473)

cot (θ) = (5.8473)

6=2+5arctan(x/2)

6 = 2 + 5 arctan (\frac{x}{2})

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(2x)+6sin(x)-2cos(x)=3 ?
The general solution for 2sin(2x)+6sin(x)-2cos(x)=3 is x= pi/6+2pin,x=(5pi)/6+2pin